Equivalent rectangular bandwidth

Researchers have derived a differential equation giving the ERB value <math>v</math> in Hz of a human auditory filter with a center frequency of <math>f</math> kHz

<math>\frac{df}{dv} = 6.23 \cdot f^2 + 93.39 \cdot f + 28.52</math>

Solving this equation gives roughly the following relation between the ERB value <math>v</math> and the frequency <math>f</math> in Hz:

<math>v = 11.17268 \cdot \log\left(1 + \frac{46.06538 \cdot f}{f + 14678.49}\right)</math>

<math>f = \frac{676170.4}{47.06538 - e^{0.08950404 \cdot v}} - 14678.49</math>